Question: If $ 2^{x} \cdot 8^{x-1} = 64 $, what is the value of $ x $?
["Solving the Equation: If $ 2^{x} \cdot 8^{x-1} = 64 $, What Is the Value of $ x $?", "When faced with an equation like $ 2^{x} \cdot 8^{x-1} = 64 $, it’s natural to wonder how to simplify and solve for $ x $. This particular exponential equation combines powers of 2 and 8—both of which relate easily to base 2—making it a perfect candidate for substitution and simplification using algebraic techniques. Understanding how to approach such problems not only strengthens algebraic skills but also builds confidence for more advanced math topics.", "### Simplifying the Equation Using Base 2", "The first step is to rewrite all terms using base 2. Since $ 8 = 2^3 $ and $ 64 = 2^6 $, we rewrite the equation as follows:", "$$\n2^{x} \cdot (2^{3})^{x-1} = 2^{6}\n$$", "Using the exponent rule $ (a^m)^n = a^{mn} $, simplify the second term:", "$$\n2^{x} \cdot 2^{3(x-1)} = 2^{6}\n$$", "Now apply the rule $ a^m \cdot a^n = a^{m+n} $:", "$$\n2^{x + 3(x - 1)} = 2^{6}\n$$", "Simplify the exponent on the left:", "$$\nx + 3x - 3 = 4x - 3\n$$", "So the equation becomes:", "$$\n2^{4x - 3} = 2^{6}\n$$", "### Setting the Exponents Equal", "Since the bases are the same, we can equate the exponents:", "$$\n4x - 3 = 6\n$$", "Solve for $ x $:", "$$\n4x = 6 + 3 = 9 \quad \Rightarrow \quad x = \frac{9}{4}\n$$", "### Final Answer", "$$\n\boxed{x = \frac{9}{4}}\n$$", "### Why This Problem Matters", "This equation demonstrates the power of expression simplification and the importance of expressing all terms with the same base. Mastering such techniques is essential not only for algebra but also for calculus, logarithms, and real-world applications involving exponential growth or decay.", "Never overlook the chance to rewrite terms—transforming $ 8^{x-1} $ into $ 2^{3(x-1)} $ was the key breakthrough that made solving for $ x $ straightforward.", "---", "Key SEO Keywords: \nSolve exponential equation #How to solve $ 2^x \cdot 8^{x-1} = 64 $ #Value of x in exponents #Algebra equations #Math tutorial #Exponential equation solved #Math problem solving", "Meta Description:\nLearn how to solve $ 2^{x} \cdot 8^{x-1} = 64 $ step by step. Discover the correct value of $ x = \frac{9}{4} $ using base conversion and exponent rules. Perfect for algebra students."]
